TITLE: Is $\{\varnothing\}\in\{\varnothing,\{\varnothing\}\}$ true?
QUESTION [15 upvotes]: Is $\{\varnothing\}\in\{\varnothing,\{\varnothing\}\}$ true? 
Can we use the same explanation as in $\{\varnothing\}\subseteq\{\varnothing,\{\varnothing\}\}$? That is, $\{\varnothing,\{\varnothing\}\}$ has two elements $\varnothing$ and $\{\varnothing\}$. The right set $\{\varnothing\}$ is its subset because it contains only one of them. Therefore, true.
With the only difference being that we replace the word subset and instead say that the right set $\{\varnothing\}$ is an element of $\{\varnothing,\{\varnothing\}\}$, therefore true?
Can we say the same thing for $\{\varnothing\}\in\{\varnothing,\{\{\varnothing\}\}\}$, and $\{\{\varnothing\}\}\in\{\varnothing,\{\varnothing\}\}$?

REPLY [4 votes]: The easiest way to resolve some of your queries is by introducing convenient notation so as not to get lost in the curly braces, such as $0=\emptyset$, $1=\{\emptyset\}$, $2=\{\emptyset,\{\emptyset\}\}=\{0,1\}$, etc. For example, the answer to your last question is negative because it is not true that $\{1\}\in\{0,1\}$, only that $\{1\}\subset\{0,1\}$.